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\title{《基础复分析》第8章级数与乘积展开 - 习题}
\author{CGZ ET AL}

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%## 《基础复分析》 习题八

\begin{enumerate}

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\item % 1

对 $\ln(1 + z/n)$ 的一个解析单值分支应用 Taylor 定理, 证明
    $$
    \lim_{n \to \infty} \left(1 + \frac{z}{n}\right)^n = e^z
    $$
    在所有紧集上是一致的.
    

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\item % 2

证明级数
    $$
    \zeta(z) = \sum_{n=1}^{\infty} n^{-z}
    $$
    在 $\operatorname{Re} z > 1$ 时收敛, 并将其导数表示为级数形式.
    

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\item % 3

证明
    $$
    (1 - 2^{1-z}) \zeta(z) = 1^{-z} - 2^{-z} + 3^{-z} - \cdots.
    $$
    并证明右边的级数当 $\operatorname{Re} z > 0$ 时是 $z$ 的解析函数.
    

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\item % 4

证明
    $$
    \sum_{n=1}^{\infty} \frac{nz^n}{1 - z^n} = \sum_{n=1}^{\infty} \frac{z^n}{(1 - z^n)^2}
    $$
    对 $|z| < 1$ 成立. (提示: 展开成二重级数并交换求和顺序.)
    

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\item % 5

将 $1/(1 + z^2)$ 展开为 $z-a$ 的幂级数, 其中 $a$ 为实数, 求系数的通项公式.
    

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\item % 6

Legendre 多项式定义为下述展开式中的系数 $P_n(\alpha)$:
    $$
    (1 - 2\alpha z + z^2)^{-1/2} = 1 + P_1(\alpha) z + P_2(\alpha) z^2 + \cdots.
    $$
    试求 $P_1, P_2, P_3, P_4$.
    

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\item % 7

试将 $\ln(\sin z/z)$ 展开为 $z$ 的幂级数, 写到前 6 项.
    

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\item % 8

Fibonacci (斐波那契) 数列定义为 $c_0 = 0$, $c_1 = 1$,
    $$
    c_n = c_{n-1} + c_{n-2}.
    $$
    证明 $c_n$ 是一个有理函数的 Taylor 展开式的系数, 并求出这个有理函数.
    

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\item % 9

函数
    $$
    S_f(z) = \frac{f'''(z)}{f'(z)} - \frac{3}{2} \left(\frac{f''(z)}{f'(z)}\right)^2
    $$
    称为 $f(z)$ 的 Schwarz 导数. 求 $S_f$ 在 $f(z)$ 的重零点或者重极点处的 Laurent 展开式的首项.
    

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\item % 10

试以 Bernoulli 数表示 $\tan z$ 的 Taylor 展开式以及 $\cot z$ 的 Laurent 展开式, 并由此证明
    $$
    \frac{\pi}{4} = 1 - \frac{1}{3} + \frac{1}{5} - \frac{1}{7} + \cdots.
    $$
    

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\item % 11

求 $1/\cos(\pi z)$ 的部分分式展开式, 并由此证明
    $$
    \frac{\pi}{4} = 1 - \frac{1}{3} + \frac{1}{5} - \frac{1}{7} + \cdots.
    $$

    

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\item % 12

求
    $$
    \sum_{n=-\infty}^{\infty} \frac{1}{(z+n)^2 + a^2}.
    $$
    

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\item % 13

证明
    $$
    \prod_{n=2}^{\infty} \left(1 - \frac{1}{n^2}\right) = \frac{1}{2}.
    $$
    

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\item % 14

证明对 $|z| < 1$, 有
    $$
    (1+z)(1+z^2)(1+z^4)\cdots = \frac{1}{1-z}.
    $$
    

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\item % 15

证明
    $$
    \prod_{n=1}^{\infty} \left(1 + \frac{z}{n}\right) e^{-\frac{z}{n}}
    $$
    在任意紧集上绝对一致收敛.
    

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\item % 16

证明绝对收敛的无穷乘积在因子重新排列后值不变.
    

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\item % 17

证明函数
    $$
    \theta(z) = \prod_{n=1}^{\infty} (1 + h^{2n-1})(1 + h^{2n-1} e^{-z})
    $$
    在全平面解析, 并满足函数方程 $\theta(z + 2\ln h) = h^{-1} e^{-z} \theta(z)$, 其中 $|h| < 1$.
    

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\item % 18

设 $\{a_n\}$ 为由互不相同的复数组成的趋于 $\infty$ 的序列, $\{A_n\}$ 为任意复数序列. 证明存在整函数 $f(z)$ 使得 $f(a_n) = A_n$.
    (提示: 设 $g(z)$ 是以 $a_n$ 为单零点的函数, 证明存在复数序列 $\{\gamma_n\}$, 使得级数
    $$
    \sum_{n=1}^{\infty} g(z) \frac{e^{\gamma_n(z-a_n)}}{z-a_n} \cdot \frac{A_n}{g'(a_n)}
    $$
    收敛.)
    

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\item % 19

证明
    $$
    \sin(z+\alpha) = e^{\pi z \cot(\pi \alpha)} \prod_{-\infty}^{\infty} \left(1 + \frac{z}{n+\alpha}\right) e^{-\pi \frac{z}{n+\alpha}},
    $$
    只要 $\alpha$ 不是整数. (提示: 将典范乘积前的因子记为 $g(z)$, 然后确定 $g'(z)/g(z)$.)
    

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\item % 20

设 $f(z)$ 是亏格为 $0$ 或者 $1$ 而且具有实零点的函数. 如果 $f(z)$ 在实轴上取实值, 证明 $f'(z)$ 的所有零点也是实数. (提示: 考虑 $\operatorname{Im}(f'(z)/f(z))$.)
    

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\item % 21

证明 Gauss 公式:
    $$
    (2\pi)^{\frac{n-1}{2}} \Gamma(z) = n^{z-\frac{1}{2}} \Gamma\left(\frac{z}{n}\right) \Gamma\left(\frac{z+1}{n}\right) \cdots \Gamma\left(\frac{z+n-1}{n}\right).
    $$
    

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\item % 22

证明:
    $$
    \Gamma\left(\frac{1}{6}\right) = 2^{-\frac{1}{3}} \left(\frac{3}{\pi}\right)^{\frac{1}{2}} \Gamma\left(\frac{1}{3}\right)^2.
    $$
    

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\item % 23

求 $\Gamma(z)$ 在极点 $z = -n$ 处的留数.
    

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\item % 24

利用 Cauchy 定理与 $\Gamma(z)$ 的积分表示计算 Fresnel (菲涅耳) 积分
    $$
    \int_0^{+\infty} \sin(x^2) \, dx, \quad \int_0^{+\infty} \cos(x^2) \, dx.
    $$
    (提示: 由 $\Gamma(z)$ 的积分表示可以计算如下概率积分:
    $$
    \int_0^{+\infty} e^{-t^2} \, dt = \frac{1}{2} \int_0^{+\infty} e^{-x} x^{-\frac{1}{2}} \, dx = \frac{1}{2} \Gamma\left(\frac{1}{2}\right) = \frac{1}{2} \sqrt{\pi}.)
    $$
    

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\item % 25

证明具有正实部的解析函数族是正规族.
    

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\item % 26

证明 $\{z^n\}$ ($n$ 为非负整数) 在单位圆盘内是正规族. 在单位圆盘外是球面度量意义下的正规族. 但是在包含单位圆周上一点的任意邻域内不正规.
    

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\item % 27

设 $f(z)$ 是整函数. 证明 $\{f(kz)\}$ ($k$ 为实数) 组成的函数族在圆环 $r_1 < |z| < r_2$ 正规, 当且仅当 $f(z)$ 是多项式.
    

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\item % 28

如果解析函数 (或者亚纯函数) 族 $\mathcal{F}$ 在区域 $\Omega$ 内不是正规的, 则存在 $z_0 \in \Omega$, 使得 $\mathcal{F}$ 在 $z_0$ 的任意一个邻域内都不是正规的.




\end{enumerate}

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